Let V be an inner product space. Let be a complete orthonormal sequence in V. Suppose that with and . Prove that .
My proof so far:
I definitely don't think I can pull that sum out in the first step because the has the sum with it as well. I was thinking of trying the following:
I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
I definitely think that I cannot just pull out the sum like I did in the top, so I was trying to use partial sums, but I don't know where to go from there. Yes, what I mean was pulling out the limits like you have, but I don't even know if that would be helpful.
There is no problem about pulling the sums out of the inner products, because the inner product is a continuous function of its two arguments. If then is the limit of the finite sums . It follows that
and similarly for the right-hand term in the inner product.